Breaking boundaries: extending the orbit-averaged Fokker-Planck equation inside the loss cone

TL;DR

This work addresses limitations in traditional loss-cone theory for tidal disruption events by introducing a sink-term into the Boltzmann equation, enabling an orbit-averaged (OAFPE) treatment that naturally yields a reaction-diffusion description of relaxation and disruptions. The authors derive the OAFPE with a loss-cone sink term, exposing a continuous framework that can reproduce standard boundary-condition results while accommodating physics operating on timescales shorter than two-body relaxation. In the steady-state, nearly radial limit they obtain an analytic interior solution inside the loss cone, including a closed-form expression for the loss-cone flux and the pericentre-related distributions, and they systematically compare this orbit-averaged approach to conventional local FP results. The proposed framework provides a versatile tool for incorporating processes such as strong scatterings, gravitational waves, and partial disruptions, potentially explaining observed TDE rates and enabling more realistic, multi-component models of galactic nuclei.

Abstract

In this Letter, we present a new formulation of loss cone theory as a reaction-diffusion system, which accounts for loss cone events through a sink term and can be orbit-averaged. It can recover the standard approach based on boundary conditions, and is derived from a simple physical model that overcomes many of the classical theoretical constraints. We test our formulation by computing the relaxed distribution of disruptive orbits in phase space, that has a simple analytic form and agrees with the pericentre of tidal disruption events at disruption predicted by non-averaged models. This formulation of the problem is particularly suitable for including more physics in tidal disruptions and the analogous problem of gravitational captures, e.g. strong scatterings, gravitational waves emission, physical stellar collisions, and repeating partial disruptions -- that can all act on timescale shorter than two-body relaxation and might cause the tension between the observed vs theoretically predicted population of tidal disruptions.

Figures (3)

• Figure 1: Comparison between the solution of the local Fokker-Planck equation inside the loss cone (numeric, blue) with the analytic limit presented in merrittDynamicsEvolutionGalactic2013 ($\mathcal{R}_\mathrm{max} = \mathcal{R}_\mathrm{LC}$, orange) and our orbit-averaged estimate (orbit-averaged, Eq. \ref{['eq:inside']}, green). On the left, we show the average distribution inside the loss cone $f_\mathrm{avg}$, on the right the distribution of $\mathcal{R}/\mathcal{R}_\mathrm{LC}$ at capture $f_\mathrm{cap}$ for stars on orbits with different $q$. For very eccentric Keplerian orbits $\mathcal{R}/\mathcal{R}_\mathrm{LC} \simeq 1/\beta$, and the distribution of penetration factor of TDEs at a given $q$ (Eq. \ref{['eq:peri_distr']}) is given by $f_\mathrm{capt} / f_\mathrm{LC}= \mathcal{K}\,\beta^2\, dN_\mathrm{capt}/d\beta$ where $\mathcal{K}$ depends on normalisation. We consider three values of the loss cone diffusivity parameter (Eq. \ref{['eq:q']}) $q=0.1$ (first row), $q=1.0$ (second row), and $q=10.0$ (third row). Each curve is normalised to its value at $\mathcal{R}_\mathrm{LC}$.
• Figure 2: Numerical solution inside the loss cone for $\tau^\mathrm{CK} = 0.005$ (solid), $\tau^\mathrm{CK} = 0.5$ (dot) and $\tau^\mathrm{CK}=0.995$ (dash) for $q=0.1$ (blue), $q=1.0$ (orange) and $q=10.0$ (green). The vertical lines mark the loss cone value $\mathcal{R}_\mathrm{LC}$. As $q$ increases, we increase $\mathcal{R}_\mathrm{max}$ according to Eq. \ref{['eq:ymax']}, as close to $\mathcal{R}_\mathrm{max}$ the distribution becomes uniform in $\tau$.
• Figure 3: Value of $\eta = \log \mathcal{R}_\mathrm{LC} / \mathcal{R}_0$ from the numerical solution ($\eta^\mathrm{num}$, blue), the fit by cohnStellarDistributionBlack1978 ($\eta^\mathrm{CK}$, Eq. \ref{['eq:eta_CK']}, orange), the prescription by merrittDynamicsEvolutionGalactic2013 ($\eta^\mathrm{M}$, Eq. \ref{['eq:eta_M']}, green), the simplified expression ($\eta^\mathrm{P}$, Eq. \ref{['eq:eta_P']}, pink) and the orbit-averaged estimate ($\eta^\mathrm{OA}$, Eq. \ref{['eq:eta']}, light blue).